Relative Nielsen Numbers, Braids and Periodic Segments

Abstract

The aim of this paper is to establish a connection between the method of period segments and the relative Nielsen fixed point theory. We prove that if W is a periodic segment over [0,T]{[0,T]} for the T-periodic semi-process Φ, then the Poincaré map P has at least N⁢(mW,W0∖W0--¯){N(m_{W},\overline{W_{0}\setminus W_{0}^{--}})} fixed points with trajectories contained in W, where N⁢(mW,W0∖W0--¯){N(m_{W},\overline{W_{0}\setminus W_{0}^{--}})} is the relative Nielsen number defined by Zhao. It is also shown that if the sequence N⁢(m¯n){N(\overline{m}^{n})} is bounded and N∞⁢(m)>1{N^{\infty}(m)>1}, then the Poincaré map has infinitely many periodic points. We prove that there exists a compact set I⊂W0{I\subset W_{0}}, invariant for the Poincaré map, such that the topological entropy h⁢(P|I){h(P|_{I})} is bounded from below by log⁡N∞⁢(m)-h⁢(m¯){\log N^{\infty}(m)-h(\overline{m})}. In particular, if h⁢(m¯)=0{h(\overline{m})=0}, then h⁢(P|I)≥log⁡N∞⁢(m).{h(P|_{I})\geq\log N^{\infty}(m).} We adapt the result obtained by Jiang to get a concrete example of a braid-like periodic segment with N∞⁢(m)>1{N^{\infty}(m)>1}.

Keywords

• nielsen number
• lefschetz number
• periodic segments
• braid group
• topological entropy
• 54h20
• 37b40
• 20f36